Daily Archives: January 5, 2011

The question mark in the title is not rhetorical. As a result of the recent discussion of free will on these pages, I’ve come to realize that I don’t really have a very clear understanding of this very basic concept — which is a big problem because my argument (that free will is inconsistent with both determinism and indeterminism) doesn’t mean anything if I can’t say what determinism is.

Here are some disjointed notes, which I hope will eventually congeal into something useful.

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The problem is that determinism and indeterminism are theories about what could happen, but the only data we have is about what actually did happen. (Actually, we don’t even have that — it has to be inferred — but let’s simplify and say we do.) It’s like looking at a single game of chess, having no previous knowledge of the rules of the game, and trying to figure out what moves the players could have made at each point in the game and whether or not they could have made different moves than they did in fact make. Of course, a single game contains only a very limited subset of all possible chess moves, and any given game has infinitely many different sets of rules which are equally consistent with it. How can you distinguish moves which are forbidden by the rules of chess from moves which are legal but which the players happened not to make in this particular game? If neither player happened to castle or to capture en passant, how could you infer that such moves were possible? If in this particular game, no pawn advanced two spaces except on its first move and no queen moved diagonally except on its first move, how could you conclude that this is a rule for the pawns but only a coincidence for the queens?

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We can conceptualize a universe as an ordered series of numbers, with each value representing a particular state of the universe and consecutive members of the series representing consecutive points in time. (Yes, we are simplifying rather drastically by making time finitely divisible and ignoring relativity, but you have to start somewhere.) Which such series can be called deterministic?

A starting definition might be that a series is deterministic if it is possible to derive the whole series by applying an algorithm to a subset thereof.

The series <1, 2, 4, 8, 16, 32, 64, 128> is deterministic by this definition. Choose any one member of the series, apply the algorithm, and you can derive the whole series.

The series <5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34> is also deterministic. Here, if you know any two members of the series, you can derive all the others by a simple algorithm.

But what about <92, 75, 30, 92, 46, 87, 69, 89, 2, 48, 630>? The first ten numbers were given by a random number generator and follow no pattern; the 11th and final number is the sum of the others. We would intuitively hesitate to call it a deterministic series, but it differs from the previous two only in degree; if you know all but one of the members, the remaining member (whichever it may be) can be derived by an algorithm. Are there perhaps degrees of determinism — the less you have to know in order to derive the whole series, the more deterministic it is?

And what about <92, 75, 30, 92, 46, 87, 69, 89, 2, 48>? Yes, that’s the same series of random number as before, minus the checksum. There’s no pattern to be found. But couldn’t we still devise a complicated ad hoc algorithm by which the whole series could be derived from one or a few of its members? Isn’t that always possible? Of course, then all the information would be in the algorithm, not in the members from which we are supposedly deriving the set — but where do we draw the line between “real” determinism and this bogus ad hoc variety?

It’s beginning to look as if determinism is a question of data compression: if lossless compression is possible, the series is deterministic. Unfortunately, I don’t know anything about data compression.

For an infinite series of numbers, the question of determinism is clearer. Since the ad hoc algorithm for an infinite series of random numbers would itself have to be infinitely long, we can say that an infinite series is deterministic if it can be derived from a finite subset of its members by a finite algorithm.

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In the examples of deterministic series given above, the whole series can be derived from any adequate subset, whether it comes from the beginning, middle, or end of the series.

But in the real universe, people (trained physicists excluded) generally believe that determinism is directional, that it runs in only one temporal direction. Which direction that is depends on who you ask and how you phrase the question. Most people will tell you that causes precede effects, that the past determines the future in a way that the future does not determine the past. Most people will also tell you what seems like the exact opposite: that present evidence can be used to reconstruct the past with a much higher degree of certainty than that with which it can be used to predict the future. Most people — and I am most definitely including myself in this group — obviously don’t know what the hell they’re talking about when it comes to determinism.

Conway’s Game of Life is a good example of a system with unidirectional determinism. Given a snapshot, a simple algorithm can tell you what will come next, yielding perfectly accurate predictions of the arbitrarily distant future — but nothing can tell you what came before.